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Article(id=1239165052431953983, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239165048212476136, articleNumber=null, orderNo=null, doi=10.12347/j.ycyk.20241121001, pmid=null, cstr=32406.14.ycyk.20241121001, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1732118400000, receivedDateStr=2024-11-21, revisedDate=1736697600000, revisedDateStr=2025-01-13, acceptedDate=null, acceptedDateStr=null, onlineDate=1773369571072, onlineDateStr=2026-03-13, pubDate=1757865600000, pubDateStr=2025-09-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773369571072, onlineIssueDateStr=2026-03-13, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773369571072, creator=13701087609, updateTime=1773369571072, updator=13701087609, issue=Issue{id=1239165048212476136, tenantId=1146029695717560320, journalId=1238841944844054536, year='2025', volume='46', issue='5', pageStart='1', pageEnd='158', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773369570067, creator=13701087609, updateTime=1773617788336, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1240206151170707811, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239165048212476136, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1240206151170707812, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239165048212476136, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=98, endPage=108, ext={EN=ArticleExt(id=1239165052671029313, articleId=1239165052431953983, tenantId=1146029695717560320, journalId=1238841944844054536, language=EN, title=Preliminary Assessment of the Doppler Positioning Performance of Low-Orbit Satellites, columnId=1239165052184482030, journalTitle=Journal of Telemetry, Tracking and Command, columnName=Navigation Technology Column, runingTitle=null, highlight=null, articleAbstract=

To evaluate the Doppler positioning performance of Low Earth Orbit (LEO) satellites, this paper analyzes the related errors and positioning performance of single-LEO navigation test satellite. Furthermore, the worldwide constellation Doppler positioning performance is analyzed in conjunction with low-orbit constellation simulation extrapolations. The results show that: ① The Doppler measurement error accuracy is at the decimetre level, which is more than one order of magnitude greater than the other error terms. Furthermore, the Doppler User Equivalent Range Rate Error (UERRE) accuracy of the comprehensive related error term is better than 0.27 m/s; ② The single-satellite Doppler positioning 3D error converges to 200 m in approximately eight minutes, with a post-convergence positioning accuracy of approximately 85 m. Furthermore, the single-star Doppler-equivalent PDOP eventually converges to around 200; ③ When the cut-off altitude angle is 10° or less and the cumulative observation time is 8 min or more, the global average of the constellation Doppler equivalent PDOP is superior to 28.8 m, the RMS is superior to 58.8 m, and better than 156.1 m on the 95%. Furthermore, the constellation Doppler positional accuracy (3D, 1σ) is superior to 7.8 m on average globally, and better than 15.9 m on the RMS, and better than 42.2 m on the 95%. The constellation Doppler-equivalent PDOP and positional accuracy are optimal at high latitudes, suboptimal at midlatitudes, and relatively poor at low latitudes.

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为了研究评估低轨卫星多普勒定位性能，本文基于实测数据评估分析了低轨卫星单星多普勒定位相关误差和定位性能，并结合低轨星座仿真推演分析了全球范围内的星座多普勒定位性能。结果表明：①在低轨试验单星多普勒定位相关误差项中，多普勒测量误差精度在分米级，大于其它误差项一个数量级以上，综合相关误差项的多普勒用户等效测速误差（UERRE）精度优于0.27 m/s；②低轨试验单星多普勒定位三维误差约在8 min收敛至200 m，收敛后定位精度（RMS）约85 m，单星多普勒等效PDOP（位置精度衰减因子）最终收敛至200左右；③在全球范围内，当截止高度角在10°及以下且累计观测8 min及以上时，星座多普勒等效PDOP全球均值优于28.8，RMS优于58.8，95%优于156.1，星座多普勒定位精度（三维，1σ）全球均值优于7.8 m，RMS优于15.9 m，95%优于42.2 m。其中，高纬度地区的星座多普勒等效PDOP和定位精度最优，中纬度地区次之，低纬度地区相对最差。

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房成贺　1992年生，硕士，高级工程师。

崔轶伦　1997年生，硕士，助理工程师。

耿长江　1984年生，博士，研究员。

宋晓丽　1986年生，硕士，高级工程师。

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房成贺　1992年生，硕士，高级工程师。

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房成贺　1992年生，硕士，高级工程师。

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崔轶伦　1997年生，硕士，助理工程师。

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崔轶伦　1997年生，硕士，助理工程师。

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2中国卫星导航系统管理办公室测试评估研究中心 北京 100094, bio={"content":"

耿长江　1984年生，博士，研究员。

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耿长江　1984年生，博士，研究员。

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2中国卫星导航系统管理办公室测试评估研究中心 北京 100094, bio={"content":"

宋晓丽　1986年生，硕士，高级工程师。

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宋晓丽　1986年生，硕士，高级工程师。

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figureFileBig=mKNrBxpSqQXuavX+s5u85Q==, tableContent=null), ArticleFig(id=1239265693275845220, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=图7, caption=低轨星座瞬时可见卫星数全球分布, figureFileSmall=u2MDZhtzmudRF5tOondazQ==, figureFileBig=mKNrBxpSqQXuavX+s5u85Q==, tableContent=null), ArticleFig(id=1239265693376508520, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=EN, label=Fig. 8, caption=Global distribution of constellation Doppler-equivalent PDOP for 8-minute cumulative observations, figureFileSmall=bg5bMikxNpG86GzTrWIIJA==, figureFileBig=8THm4zSHkYrQtz4olPC7Cg==, tableContent=null), ArticleFig(id=1239265693464588908, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=图8, caption=8 min累计观测的星座多普勒等效PDOP全球分布, figureFileSmall=bg5bMikxNpG86GzTrWIIJA==, figureFileBig=8THm4zSHkYrQtz4olPC7Cg==, tableContent=null), 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tableContent=null), ArticleFig(id=1239265693867242116, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=图10, caption=15 min累计观测的星座多普勒等效PDOP全球分布, figureFileSmall=ZdTLTzOCe8p4MYK0kan+DA==, figureFileBig=RI0mZizu8cI5FLB6ZOKAmQ==, tableContent=null), ArticleFig(id=1239265693963711112, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=EN, label=Table 1, caption=

Statistical results of the analysis of error terms associated with single LEO satellite Doppler positioning

, figureFileSmall=null, figureFileBig=null, tableContent=
误差项A站--SY-01星B站--SY-02星
STDRMSSTDRMS
多普勒测量误差0.260.230.230.23
电离层延迟变化率0.030.030.020.03
对流层延迟变化率0.030.030.030.03
相对论效应引起的延迟变化率0.000 30.000 40.000 20.000 8
Sagnac效应引起的延迟变化率0.000 10.000 70.000 40.001 5
卫星钟漂0.0020.0440.0010.041
), ArticleFig(id=1239265694060180109, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=表1, caption=

低轨多普勒定位相关各项误差统计结果

, figureFileSmall=null, figureFileBig=null, tableContent=
误差项A站--SY-01星B站--SY-02星
STDRMSSTDRMS
多普勒测量误差0.260.230.230.23
电离层延迟变化率0.030.030.020.03
对流层延迟变化率0.030.030.030.03
相对论效应引起的延迟变化率0.000 30.000 40.000 20.000 8
Sagnac效应引起的延迟变化率0.000 10.000 70.000 40.001 5
卫星钟漂0.0020.0440.0010.041
), ArticleFig(id=1239265694131483280, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=EN, label=Table 2, caption=

Average number of visible LEO satellites in different regions of the globe

, figureFileSmall=null, figureFileBig=null, tableContent=
截止高度角高纬地区(南北纬60°以上)中纬地区(南北纬30°~60°)低纬地区(南北纬30°以内)全球
5°8.23.62.44.8
10°6.82.61.73.8
15°5.51.91.32.9
), ArticleFig(id=1239265694202786451, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=表2, caption=

全球不同区域内的平均可见低轨卫星数

, figureFileSmall=null, figureFileBig=null, tableContent=
截止高度角高纬地区(南北纬60°以上)中纬地区(南北纬30°~60°)低纬地区(南北纬30°以内)全球
5°8.23.62.44.8
10°6.82.61.73.8
15°5.51.91.32.9
), ArticleFig(id=1239265694269895318, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=EN, label=Table 3, caption=

Statistical results of Doppler-equivalent PDOP for LEO constellations in different regions of the globe

, figureFileSmall=null, figureFileBig=null, tableContent=
区域截止高度角8 min多普勒等效PDOP10 min多普勒等效PDOP15 min多普勒等效PDOP
平均RMS95%平均RMS95%平均RMS95%
高纬地区5°5.045.167.204.484.586.253.673.755.20
10°5.485.678.294.855.007.233.974.106.03
15°6.136.4610.055.385.628.594.414.617.04
中纬地区5°10.2110.4814.128.889.0711.987.287.449.72
10°17.6535.1325.3213.2419.9721.539.6711.4416.73
15°86.23593.30274.6927.7548.88127.0515.3120.4753.23
低纬地区5°25.1938.9695.1819.8526.3360.9513.6715.5131.35
10°64.2396.15218.5840.4754.17113.1922.6126.3845.96
15°274.551 237.26658.5087.66115.05241.5136.9642.8177.47
全球5°13.6023.3338.1410.9816.1929.918.1410.1120.20
10°28.7958.79156.0419.3233.2392.2211.9716.6641.53
15°120.71787.78485.8439.7971.75181.1218.6927.3358.11
), ArticleFig(id=1239265694378947226, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=表3, caption=

全球不同区域内的低轨星座多普勒等效PDOP统计结果

, figureFileSmall=null, figureFileBig=null, tableContent=
区域截止高度角8 min多普勒等效PDOP10 min多普勒等效PDOP15 min多普勒等效PDOP
平均RMS95%平均RMS95%平均RMS95%
高纬地区5°5.045.167.204.484.586.253.673.755.20
10°5.485.678.294.855.007.233.974.106.03
15°6.136.4610.055.385.628.594.414.617.04
中纬地区5°10.2110.4814.128.889.0711.987.287.449.72
10°17.6535.1325.3213.2419.9721.539.6711.4416.73
15°86.23593.30274.6927.7548.88127.0515.3120.4753.23
低纬地区5°25.1938.9695.1819.8526.3360.9513.6715.5131.35
10°64.2396.15218.5840.4754.17113.1922.6126.3845.96
15°274.551 237.26658.5087.66115.05241.5136.9642.8177.47
全球5°13.6023.3338.1410.9816.1929.918.1410.1120.20
10°28.7958.79156.0419.3233.2392.2211.9716.6641.53
15°120.71787.78485.8439.7971.75181.1218.6927.3358.11
), ArticleFig(id=1239265694450250397, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=EN, label=Table 4, caption=

Doppler positioning accuracy derivation results for LEO constellations in different regions of the globe

, figureFileSmall=null, figureFileBig=null, tableContent=
区域截止8 min多普勒定位10 min多普勒定位15 min多普勒定位
高度角平均RMS95%平均RMS95%平均RMS95%
高纬地区5°1.361.391.941.211.241.690.991.011.40
10°1.481.532.241.311.351.951.071.111.63
15°1.661.742.711.451.522.321.191.241.90
中纬地区5°2.762.833.812.402.453.231.972.012.62
10°4.779.496.843.575.395.812.613.094.52
15°23.28160.1974.177.4913.2034.304.135.5314.37
低纬地区5°6.8010.5225.705.367.1116.463.694.198.46
10°17.3425.9659.0210.9314.6330.566.107.1212.41
15°74.13334.06177.8023.6731.0665.219.9811.5620.92
全球5°3.676.3010.302.964.378.082.202.735.45
10°7.7715.8742.135.228.9724.903.234.5011.21
15°32.59212.70131.1810.7419.3748.905.057.3815.69
), ArticleFig(id=1239265694567690915, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239165052431953983, language=CN, label=表4, caption=

全球不同区域内的低轨星座多普勒定位精度推演结果

, figureFileSmall=null, figureFileBig=null, tableContent=
区域截止8 min多普勒定位10 min多普勒定位15 min多普勒定位
高度角平均RMS95%平均RMS95%平均RMS95%
高纬地区5°1.361.391.941.211.241.690.991.011.40
10°1.481.532.241.311.351.951.071.111.63
15°1.661.742.711.451.522.321.191.241.90
中纬地区5°2.762.833.812.402.453.231.972.012.62
10°4.779.496.843.575.395.812.613.094.52
15°23.28160.1974.177.4913.2034.304.135.5314.37
低纬地区5°6.8010.5225.705.367.1116.463.694.198.46
10°17.3425.9659.0210.9314.6330.566.107.1212.41
15°74.13334.06177.8023.6731.0665.219.9811.5620.92
全球5°3.676.3010.302.964.378.082.202.735.45
10°7.7715.8742.135.228.9724.903.234.5011.21
15°32.59212.70131.1810.7419.3748.905.057.3815.69
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低轨卫星多普勒定位性能初步评估
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房成贺 1, 2 , 崔轶伦 1, 2 , 耿长江 1, 2 , 宋晓丽 1, 2
遥测遥控 | 导航技术专栏 2025,46(5): 98-108
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遥测遥控 | 导航技术专栏 2025, 46(5): 98-108
低轨卫星多普勒定位性能初步评估
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房成贺1, 2, 崔轶伦1, 2, 耿长江1, 2, 宋晓丽1, 2
作者信息
  • 1北京遥测技术研究所 北京 100076
  • 2中国卫星导航系统管理办公室测试评估研究中心 北京 100094

    • 房成贺　1992年生，硕士，高级工程师。

    崔轶伦　1997年生，硕士，助理工程师。

    耿长江　1984年生，博士，研究员。

    宋晓丽　1986年生，硕士，高级工程师。

Preliminary Assessment of the Doppler Positioning Performance of Low-Orbit Satellites
Chenghe FANG1, 2, Yilun CUI1, 2, Changjiang GENG1, 2, Xiaoli SONG1, 2
Affiliations
  • 1.Beijing Research Institute of Telemetry, Beijing 100076, China
  • 2.Test and Assessment Research Center of China Satellite Navigation Office, Beijing 100094, China
出版时间: 2025-09-15 doi: 10.12347/j.ycyk.20241121001
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摘要
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为了研究评估低轨卫星多普勒定位性能，本文基于实测数据评估分析了低轨卫星单星多普勒定位相关误差和定位性能，并结合低轨星座仿真推演分析了全球范围内的星座多普勒定位性能。结果表明：①在低轨试验单星多普勒定位相关误差项中，多普勒测量误差精度在分米级，大于其它误差项一个数量级以上，综合相关误差项的多普勒用户等效测速误差（UERRE）精度优于0.27 m/s；②低轨试验单星多普勒定位三维误差约在8 min收敛至200 m，收敛后定位精度（RMS）约85 m，单星多普勒等效PDOP（位置精度衰减因子）最终收敛至200左右；③在全球范围内，当截止高度角在10°及以下且累计观测8 min及以上时，星座多普勒等效PDOP全球均值优于28.8，RMS优于58.8，95%优于156.1，星座多普勒定位精度（三维，1σ）全球均值优于7.8 m，RMS优于15.9 m，95%优于42.2 m。其中，高纬度地区的星座多普勒等效PDOP和定位精度最优，中纬度地区次之，低纬度地区相对最差。

低轨卫星  /  多普勒定位  /  性能评估  /  精度因子

To evaluate the Doppler positioning performance of Low Earth Orbit (LEO) satellites, this paper analyzes the related errors and positioning performance of single-LEO navigation test satellite. Furthermore, the worldwide constellation Doppler positioning performance is analyzed in conjunction with low-orbit constellation simulation extrapolations. The results show that: ① The Doppler measurement error accuracy is at the decimetre level, which is more than one order of magnitude greater than the other error terms. Furthermore, the Doppler User Equivalent Range Rate Error (UERRE) accuracy of the comprehensive related error term is better than 0.27 m/s; ② The single-satellite Doppler positioning 3D error converges to 200 m in approximately eight minutes, with a post-convergence positioning accuracy of approximately 85 m. Furthermore, the single-star Doppler-equivalent PDOP eventually converges to around 200; ③ When the cut-off altitude angle is 10° or less and the cumulative observation time is 8 min or more, the global average of the constellation Doppler equivalent PDOP is superior to 28.8 m, the RMS is superior to 58.8 m, and better than 156.1 m on the 95%. Furthermore, the constellation Doppler positional accuracy (3D, 1σ) is superior to 7.8 m on average globally, and better than 15.9 m on the RMS, and better than 42.2 m on the 95%. The constellation Doppler-equivalent PDOP and positional accuracy are optimal at high latitudes, suboptimal at midlatitudes, and relatively poor at low latitudes.

Low-Orbit satellites  /  Doppler positioning  /  Performance assessment  /  Dilution of precision
房成贺, 崔轶伦, 耿长江, 宋晓丽. 低轨卫星多普勒定位性能初步评估. 遥测遥控, 2025 , 46 (5) : 98 -108 . DOI: 10.12347/j.ycyk.20241121001
Chenghe FANG, Yilun CUI, Changjiang GENG, Xiaoli SONG. Preliminary Assessment of the Doppler Positioning Performance of Low-Orbit Satellites[J]. Journal of Telemetry, Tracking and Command, 2025 , 46 (5) : 98 -108 . DOI: 10.12347/j.ycyk.20241121001
正文
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0　引言
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全球卫星导航系统(Global Navigation Satellite System, GNSS)信号弱、穿透能力差、易被欺骗干扰等固有弱点对定位导航授时(Positioning, Navigation and Timing, PNT)体系的建设提出挑战，寻求可互换、可替代和互补备份的PNT技术，发展综合PNT体系已成为未来时空服务发展的关键。因此，研究构建GNSS应急备份无线电导航定位系统也成为近年来的研究热点。低轨卫星具有轨道高度低、信号空间损耗小、落地功率高、抗干扰能力强等特点[1]，同时得益于低轨卫星对地运动速度快、地面接收信号的多普勒频移大且易测量的特性[2]，使得基于多普勒特性的定位技术非常适合低轨卫星定位系统，可作为GNSS的应急备份定位技术手段。
近年来随着低轨星座的快速发展，不少学者从不同方面和角度对低轨多普勒定位技术进行了详细研究分析。Psiaki等利用载波多普勒频移和伪距观测值提出了多种精度衰减因子指标，并使用仿真星座和Oneweb数据进行了性能验证[3,4]。他提出了基于多普勒测量的全球导航概念用以替代基于伪距的全球导航，经实验验证：当多普勒测量精度为0.01 m/s时定位精度可达1~5 m、测速精度可达0.01~0.05 m/s[5]。Psiaki验证了利用低轨卫星的多普勒测量数据与INS(Inertial Navigation System, 惯性导航系统)数据相结合，以实现类似全球导航卫星系统导航功能的可行性[6]。邓志鑫等为了解决用户在不同低轨卫星可见数量条件下的连续定位解算问题，提出多种不同可见星数量条件下仅利用卫星多普勒观测信息的定位解算方法[7]。郭斐等研究提出了基于一个由288颗LEO卫星构成的Walker星座，选取仿真了9个全球均匀分布的地面站多普勒观测值，建立了瞬时多普勒定位数学模型，并评估了其潜在的服务性能[8]。周舒涵等分析了低轨卫星的对地覆盖特性、信号传输特性以及多普勒频移特性，推导了多普勒定位原理和方法，并基于已在轨的铱星和全球星系统进行了定位仿真实验与分析[9]。施闯等针对在低轨运行的类GNSS导航系统，提出了一种仅利用多普勒数据的定位解算算法，该算法能够同时估计接收机时钟漂移、位置和速度，并仿真分析了多普勒定位误差和定位性能[10]。张雨露等基于典型的Starlink星座构型，对大规模低轨星座多普勒定位算法展开了研究[2]。其他国内外学者基于美国第二代低轨卫星通信系统(Iridium Next)、轨道通信(Orbcomm)、全球星(Globalstar)和星链(Starlink)星座陆续提出了低轨卫星信号的捕获和跟踪、多普勒频移测量和多普勒定位算法[11−13]。
现有研究大多集中在低轨多普勒定位的理论算法研究和仿真分析方面，尚未见基于实测数据的低轨导航卫星多普勒定位性能评估研究。因此，本文首先利用低轨卫星实测数据对低轨单星多普勒定位相关误差和定位性能进行评估分析，进而通过对低轨星座全球可见卫星数和星座多普勒等效PDOP进行仿真分析，最后结合实测获得的低轨试验单星多普勒用户等效距离变化率误差精度，对全球范围内星座多普勒定位精度进行推演分析。
1　评估基本原理
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1.1　低轨单星多普勒定位原理
多普勒定位观测方程为：
式中，表示伪距变化率，为多普勒观测值，λ为对应频率的波长，为卫星速度矢量，vr=[vrx,vry,vrz]为接收机速度矢量，l是测站到卫星的单位矢量，rs=[xs,ys,zs]为卫星位置矢量，rr=[xr,yr,zr]为接收机位置矢量，为接收机钟漂，是卫星钟漂，表示由相对论效应引起的延迟变化率，为电离层延迟变化率，为对流层延迟变化率，表示由Sagnac效应引起的延迟变化率，ε表示其它未被模型化的观测误差，待估计量为。其中，和可分别由以下公式计算：
式中，we表示地球自转角速度，c为光速。
实际评估采用静态测试，则认为测站速度为零，则：
单星多普勒定位实际上是利用卫星前后不同历元的数据进行定位的，其物理量表征了站星间的距离变化率。当用户接收到某颗卫星的信号时，可测量出信号的多普勒频移，且用户必然位于以该卫星位置为锥顶的等多普勒圆锥曲面上；该曲面上所有点的多普勒频移值相同，均等于多普勒频移测量值。当测站位置静止于地面某一点时，随着卫星过境，测站采集到的数据不断增加，在这一时段内认为卫星钟稳定，则单星多普勒定位原理如图1所示。由图可以看出，不同历元形成的圆锥面会存在交集，测站则包含于其中。
在定位解算中，由式(2)可得到第i个观测历元的对待估参数的偏导数向量ai和观测值残差(Observed Minus Computed, OMC)值bi分别为：
由于低轨卫星运行速度快，覆盖范围内不同位置用户可观测单星时间仅为几至十几分钟，在有限的观测时间内累积n(≥4)个多普勒测量值即可实现用户的单星多普勒连续观测定位解算，在式(3)和式(4)单历元观测基础上，可得n个历元的观测方程设计矩阵A和观测残差向量b如下：
根据牛顿迭代最小二乘法原理，并考虑高度角加权，则待估参数(接收机位置和接收机钟速)求解公式如下：
式中，P为观测值权矩阵。其中，第i个多普勒观测历元的权重Pi计算公式为Pi=sin2(θi)，θi为相应低轨卫星高度角。
主要计算步骤如下：
①给定初值条件rr、；
②计算观测矩阵A、观测量残差b；
③利用最小二乘法求出位置改正量Δrr和接收机钟速改正量；
④利用修正量Δrr、修正rr、，然后重复步骤②~③；
⑤如果修正量满足一定精度的需求或者迭代次数达到迭代上限则输出测站坐标rr和接收机钟漂。
通过对观测弧段内各历元的单星多普勒定位结果与测站参考坐标真值作差，并统计相应定位精度和收敛时间，评估单星多普勒定位及收敛性能。
1.2　低轨星座多普勒定位性能推演原理
目前阶段仅能同时观测一颗低轨卫星，若想得到整星座情况下静态用户多星可视条件下的多普勒定位精度水平，则可以从单星的测试结果推演整个星座的定位性能。根据试验单星可视情况，通过单星多普勒测量精度，结合星座多普勒定位精度衰减因子(Dilution of Precision, DOP)值可推导得到星座多普勒定位精度。
传统GNSS伪距定位GDOP(Geometric Dilution of Precision，几何精度衰减因子)反映了用户坐标和接收机钟差与伪距观测误差方差之间的关系。在利用多普勒定位时，观测量是多普勒频移，GDOP值反映的是用户坐标和接收机频偏与多普勒测量误差方差之间的关系[9]。
由式(6)可以推导用户位置误差(3个方向分量)和接收机频偏误差δu与多普勒频移观测误差δf之间的关系为：
式中，A为多普勒观测设计矩阵，其定义及计算过程见式(5)和式(7)。
假设接收机跟踪接收卫星信号测量得到的每个多普勒观测值之间相互独立，观测误差服从标准正态分布(均值为0、方差为)。根据方差-协方差传播定律，用户坐标和接收机频偏误差的协方差可表示为：
上式中D表示方差-协方差矩阵。定义权阵Q，表示对多普勒测量误差的放大因子：
则基于多普勒定位的GDOP表示为：
同样地，PDOP可以表示为：
由式(5)、式(7)和式(12)可以看出，矩阵Q和A仅与可见卫星的个数、位置速度以及播发的信号频率有关，与接收机的好坏、信号强度的大小并无关联。对于同一种卫星系统，播发的信号频率相同，此时GDOP和PDOP只与可见卫星的位置、速度和数量有关。GDOP与PDOP值越小，多普勒测量误差对接收机坐标和频率的解算结果影响越小，定位结果越可靠，定位精度越高[9]。因此，可以用多普勒GDOP与PDOP值来分析低轨卫星多普勒定位系统的性能，以及从单星推算到多星多普勒定位的性能。
由式(10)~(14)可得如下关系式：
上式中，多普勒pDOP可以为GDOP或PDOP(单位为s)等精度因子之一，而σUERRE, Doppler即为低轨单星多普勒用户等效距离变化率误差(User Equivalent Range Rate Error，UERRE)精度(单位为m/s)，当多普勒pDOP为PDOP时，mPos, Doppler则表示为三维多普勒定位精度(1σ，单位为m)。可见，若星座多普勒PDOP已知，并假设星座所有卫星多普勒用户等效距离变化率误差等精度均为σUERRE,Doppler，则可推演得到星座多普勒定位精度。
2　试验数据
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本文试验评估数据基于两组低轨卫星(近极轨，轨道高度约1 100 km)扩展L信号(频率约1.5 GHz)的实测数据，其中，一组为在A站(106.5°E，29.5°N)静态观测低轨卫星(假定其编号为SY-01)得到的原始多普勒测量数据，另一组为在B站(130.5°E，46.5°N)静态观测低轨卫星(假定其编号为SY-02)得到的原始多普勒测量数据，数据采样率为0.06 s。地面站坐标、低轨卫星轨道和钟差信息已知。图2所示为两组试验数据的低轨试验卫星相对地面观测站的星空轨迹方向图。
当前实际在轨低轨卫星数量较少，因此本文主要基于实测试验数据开展低轨单星多普勒定位相关误差分析和低轨单星多普勒定位性能评估，并基于实测单星多普勒测量精度和定位性能开展星座多普勒定位性能仿真推演。
需要注意的是，利用低轨卫星多普勒数据进行定位解算过程中，无法像GNSS伪距单点定位那样将地心位置作为初始值进行迭代求解。原因是低轨卫星轨道高度远小于地球半径，线性化结果误差较大，导致以地心位置为初始值的迭代不收敛[8]。因此需要给出较为准确的用户位置初值，一旦用户坐标初值与真实值距离较远(如大于100 km以上)时，很可能会导致迭代计算结果不收敛的情况[7]。
3　试验评估与结果分析
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3.1　低轨多普勒定位相关误差分析
低轨单星多普勒定位相关误差分析，主要从地面静态站跟踪低轨卫星信号方向的电离层延迟变化率、对流层延迟变化率、相对论效应引起的延迟变化率、Sagnac效应引起的延迟变化率和低轨卫星钟漂等多普勒定位误差项量级大小，以及低轨多普勒数据测量误差及精度等方面开展分析。
图3为采用多普勒历元间二次差分方法评估分析得到的上述两组低轨试验卫星多普勒观测数据的测量误差序列及精度统计结果。结果表明：A站采集的SY-01星多普勒观测数据的测量误差整体基本在±5 Hz(约等效为±1.0 m/s)内，测量精度STD(Satellite Tracking and Data, 卫星跟踪与数据)和RMS(Remote Monitoring System, 远程监控系统)统计结果均为1.34 Hz(等效为0.26 m/s)；B站采集的SY-02星多普勒观测数据的测量误差整体也基本在±5 Hz(约等效为±1.0 m/s)内，测量精度STD和RMS统计结果均为1.16 Hz(等效为0.23 m/s)。
图4为分别利用北斗电离层模型(BDGIM)和Saastamoinen+UNB3(两款对流层延迟改正模型)，并采用历元间差分法评估分析，得到两个低轨试验卫星静态观测弧段内信号传播方向上的对流层和电离层延迟变化率序列及统计结果。结果表明：A站观测SY-01星单个可视弧段内，电离层延迟变化率基本在-0.04~0.06 m/s范围内波动，相应的STD和RMS统计值均为0.03 m/s；对流层延迟变化率基本在±0.09 m/s范围内变化，相应的STD和RMS统计值也均为0.03 m/s。B站观测SY-02星得单个可视弧段内，电离层延迟变化率基本在-0.02~0.06 m/s范围内变化，相应STD和RMS统计值分别为0.02 m/s和0.03 m/s；对流层延迟变化率基本在±0.09 m/s范围内变化，相应STD和RMS统计值均为0.03 m/s。另外，可见两组试验弧段内的电离层和对流层延迟变化率基本均以最大高度角时刻对应值为中心，呈中心对称现象。
图5为利用式(2)、式(3)评估分析得到的由相对论效应、Sagnac效应引起的延迟变化率和利用低轨卫星钟差信息评估分析得到的卫星钟漂变化序列及相应统计结果。表1为上述两组试验数据的低轨多普勒定位相关各项误差(如多普勒测量误差、电离层延迟变化率、对流层延迟变化率、相对论效应引起的延迟变化率、Sagnac效应引起的延迟变化率和卫星钟漂)的STD和RMS统计结果汇总。因在低轨单星多普勒定位解算中，接收机钟漂通常采用常数估计，各项误差的系统偏差(常数部分)可被接收机钟漂参数吸收[14,15]。因此本文主要从各误差项STD统计值方面对其影响情况进行对比分析。从表中可见，多普勒测量误差影响量级最大，约在分米/秒量级；对流层延迟和电离层延迟变化率影响量级次之，约在厘米/秒量级；卫星钟漂影响量级再次之，约在毫米/秒量级；相对论效应和Saganc效应引起的延迟变化率影响量级最小，相比卫星钟漂约小一个数量级。根据上述各项误差STD精度统计结果，由方差-协方差传播定律可得两颗低轨试验卫星的实测多普勒用户等效距离变化率误差精度分别为0.263 m/s和0.233 m/s。
3.2　低轨单星多普勒定位性能评估结果
低轨单星多普勒定位性能评估具体解算策略如下：采用牛顿最小二乘迭代多普勒定位解算方法，截止高度角为10°，同时采用高度角加权。由于上述测站均配备高精度原子钟，在低轨观测弧段(最长10余分钟)内接收机钟漂参数与接收机静态位置参数均采用常数估计。
图6展示了基于单星多普勒定位算法对两组低轨试验卫星的评估结果，包括定位误差序列、定位精度及收敛时间统计。结果表明：A站观测SY-01星的单星多普勒定位误差约在7.69 min后收敛至200 m，收敛后定位精度(RMS)为83.29 m，最终多普勒PDOP收敛稳定至200 m左右；B站观测SY-02星的单星多普勒定位误差约在7.67 min后收敛至200 m，收敛后定位精度(RMS)为84.21 m，最终多普勒PDOP收敛稳定至200 m左右。
3.3　星座多普勒定位性能推演分析结果
星座多普勒定位性能仿真推演基于近极轨道星座构型，星座仿真基本参数为：卫星总数60颗(均匀分布在6个轨道面)、轨道高度1 170 km、轨道倾角86.4°、相位因子3。按照全球5°×2.5°经纬度格网划分格网点，分析全球各格网点处的星座低轨可见卫星数和星座低轨静态多历元多普勒等效PDOP的全球分布情况，其中静态多历元采样间隔为0.06 s。
图7分别给出了5°、 10°和 15°截止高度角条件下的低轨星座瞬时可见卫星数全球分布情况。可见该低轨星座在15°以下截止高度角条件下可实现全球至少1重覆盖；其中，高纬度地区可见卫星数相对较多(基本在4~9颗)，中低纬地区可见卫星数相对较少(基本在1~4颗)；随着截止高度角的增大，全球可见卫星数整体都有不同程度下降，且可见卫星数在3颗及以下的蓝色区域进一步扩大，甚至扩大到南北纬60°以上。
表2给出了5°、10°和15°截止高度角条件下高、中、低纬度地区和全球范围内的平均可见卫星数统计结果。结果表明，在上述不同截止高度角条件下，高纬地区平均可见卫星数约在5.5~8.2颗，中纬地区平均可见卫星数约在1.9~3.6颗，低纬地区平均可见卫星数约在1.3~2.4颗，全球平均可见卫星数约在2.9~4.8颗。可见中低纬地区的可见卫星数基本都在4颗以下，即表明相应区域不满足单历元瞬时多普勒定位的基本条件。因此，该低轨星座主要应用场景和模式为静态多历元累计多普勒定位模式(后文“星座多普勒定位”均默指该定位模式)。
图8、图9和图10分别给出了5°、10°和15°截止高度角条件下的全球各格网点处8 min、10 min和15 min累计观测的星座多普勒等效PDOP(即根据静态多历元连续观测计算得到的多普勒PDOP)分布。可见在相同的累计观测时长下，随着截止高度角的增大，中低纬度局部区域的星座多普勒等效PDOP存在不同程度的变差(增大)，这主要是因为随截止高度角增大，使本就相对较少的中低纬度区域的可见卫星数进一步减少，加剧了相应区域多普勒观测几何构型的恶化(即等效PDOP变差)。另外，在相同截止高度角条件下，随着累计观测时长的增加，原本星座多普勒等效PDOP较差的中低纬度局部区域也会得到明显改善提升。
表3分别给出了在5°、10°、15°截止高度角观测条件下，高、中、低纬度地区和全球范围内的各格网点处8 min、10 min和15 min累计观测的星座多普勒等效PDOP均值、RMS和95%统计结果。结果表明：在高纬度地区，当截止高度角在15°及以下且累计观测时间在8 min及以上时，区域内星座多普勒等效PDOP均值优于6.2，RMS优于6.5，95%优于10.1；在中纬度地区，当截止高度角在10°及以下且累计观测时间在8 min及以上时，区域内星座多普勒等效PDOP均值优于17.7，RMS优于35.1，95%优于25.3；在低纬度地区，当截止高度角在5°及以下且累计观测在8 min及以上时，区域内星座多普勒等效PDOP均值优于25.2，RMS优于39.0，95%优于95.2；在全球范围内，当截止高度角在10°及以下且累计观测8 min及以上时，全球星座多普勒等效PDOP均值优于28.8，RMS优于58.8，95%优于156.1。
根据前述两颗低轨试验卫星的实测数据，多普勒用户等效距离变化率误差精度σUERRE, Doppler均优于0.27 m/s ，假设星座所有卫星多普勒测量等精度相同且均为0.27 m/s，则可结合表2中的星座多普勒等效PDOP统计结果，进一步推演得到全球范围内星座多普勒理论定位精度(三维，1σ)，如表3所示。
表4分别给出了在5°、10°、15°截止高度角观测条件下，高、中、低纬度地区和全球范围内的各格网点处8 min、10 min和15 min累计观测的星座多普勒定位精度推演结果(均值、RMS和95%统计值)。推演结果表明：在高纬度地区，当截止高度角在15°及以下且累计观测在8 min及以上时，区域内星座多普勒定位精度均值优于1.7 m，RMS优于1.8 m，95%优于2.8 m；在中纬度地区，当截止高度角在10°及以下且累计观测在8 min及以上时，区域内星座多普勒定位精度均值优于4.8 m，RMS优于9.5 m，95%优于6.9 m；在低纬度地区，当截止高度角在5°及以下且累计观测在8 min及以上时，区域内星座多普勒定位精度均值优于6.8 m，RMS优于10.6 m，95%优于25.8 m；在全球范围内，当截止高度角在10°及以下且累计观测8 min及以上时，全球星座多普勒定位精度均值优于7.8 m，RMS优于15.9 m，95%优于42.2 m。
4　结束语
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针对低轨导航卫星多普勒定位性能，本文首先介绍了低轨单星多普勒定位基本原理和低轨星座多普勒定位性能推演原理，基于实测低轨试验卫星数据评估分析了低轨单星多普勒定位相关误差和定位性能，然后仿真分析了低轨星座全球可见卫星数和星座多普勒等效PDOP全球分布情况，最后基于实测多普勒用户等效距离变化率误差精度和仿真的等效PDOP推演分析了全球范围内星座多普勒定位精度。结论如下：
①低轨试验单星多普勒定位相关误差分析结果表明：多普勒测量误差影响量级最大，约在分米/秒量级；对流层延迟和电离层延迟变化率影响量级次之，约在厘米/秒量级；卫星钟漂影响量级再次之，约在毫米/秒量级；相对论效应和Saganc效应引起的延迟变化率影响量级最小，相比卫星钟漂约小一个数量级。综合上述各项误差精度，实测单星多普勒用户等效距离变化率误差精度优于0.27 m/s。
②低轨单星多普勒定位性能评估结果表明：A站观测SY-01星和B站观测SY-02星的单星多普勒定位误差约在8 min收敛至200 m，收敛后定位精度(RMS)约85 m，最终多普勒PDOP收敛稳定至200左右。
③星座多普勒定位性能推演分析结果表明：在全球范围内，当截止高度角在10°及以下且累计观测8 min及以上时，星座多普勒等效PDOP全球均值优于28.8，RMS优于58.8，95%优于156.1，星座多普勒定位精度(三维，1σ)全球均值优于7.8 m，RMS优于15.9 m，95%优于42.2 m。其中，高纬度地区的星座多普勒等效PDOP和定位精度最优，中纬度地区次之，低纬度地区相对最差。
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2025年第46卷第5期
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doi: 10.12347/j.ycyk.20241121001
  • 接收时间：2024-11-21
  • 首发时间：2026-03-13
  • 出版时间：2025-09-15
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  • 收稿日期：2024-11-21
  • 修回日期：2025-01-13
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    1北京遥测技术研究所 北京 100076
    2中国卫星导航系统管理办公室测试评估研究中心 北京 100094
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多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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